The PI proposes to investigate the long-time behavior of solutions for some classes of dispersive and hyperbolic PDEs. The proposed research revolves in particular around three classes of problems. The first concerns the question of stability of flat interfaces for the irrotational free boundary Euler equations describing the motion of water waves. The second pertains to the study of the asymptotic behavior for nonlinear systems of mixed hyperbolic/dispersive type, such as the Maxwell-Schroedinger, Klein-Gordon-Zakharov, and Wave-Schroedinger systems. The third is related to singular limits over long intervals of time for point wise decaying solutions of some nonlinear hyperbolic PDEs. The PI's general aim is to prove, under certain conditions, global existence of solutions evolving from small Cauchy data. Furthermore, he aims to characterize the behavior of solutions at infinity and, in particular, answer the question of scattering, determining whether the global solutions constructed behave or not like linear ones for large times. A central theme of the proposed research will be the identification of the mechanisms determining at highest order the global regularity of solutions and the stability of certain equilibria. The PI expects some of the questions addressed to have highly non-trivial answers regarding the qualitative behavior of solutions for large times.
The equations under consideration describe a variety of physical phenomena such as fluid flows, quantum mechanics and plasma turbulence. Their relevance is widely recognized and supported by numerical and experimental evidence. Understanding the quantitative and qualitative properties of solutions, and their behavior over long periods of time, would be of importance for several research areas in the theoretical and applied sciences. For example, proving global-in-time stability of water waves models will improve our understanding of the motion of waves, such as those generated offshore on the surface of the deep ocean. It would also be relevant to numerical investigations and experiments aiming to understand how rogue waves are generated from apparently calm waters. Establishing mathematically rigorous stability results in some specific regimes would rule out some of their possible causes. Furthermore, proving that certain specific nonlinear asymptotic behaviors can occur in some of the equations considered, would reveal deeper connections with other fundamental models such as the nonlinear Schroedinger equation, which is ubiquitous in many branches of mathematics, physics, and engineering. The PI believes that solving the proposed problems would require substantial extensions of the current analytical methods, which may result in significant technical improvements of interest for the mathematics community at large. Finally, beyond pursuing his own projects, the PI will also be actively engaged in the dissemination of his research and will strongly promote collaborations among different groups within the international PDE community and researchers in related areas.
